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Normally one would compress a message (making the message smaller, to save storage or channel bandwidth), then transform it cryptographically for secrecy (without changing the message length), and finally add bits to the message to allow for error detection or correction. 3 Channel Capacity. Shannon also introduced the concept of channel capacity, which is the maximum rate at which bits can be sent over an unreliable (noisy) information channel with arbitrarily good reliability. The channel capacity is represented as a fraction or percentage of the total rate at which bits can be sent physically over the channel.

The code uses extra redundant bits to check for errors, and performs the checks with special check equations. A parity check equation of a sequence of bits just adds the bits of the sequence and insists that the sum be even (for even parity) or odd (for odd parity). This chapter uses by and take the even parity. Alternatively, one says that the sum is taken modulo (divide ☛ remainder), or one says that the sum is taken over the integers mod , . A simple parity check will detect if there has been an error in one bit position, since even parity will change to odd parity.

We will usually be using binary codes, that is, codes that use only the binary bits and . There are three kinds of coding: ✝ ✞ 3. Coding and Information Theory 25 1. Source Coding. This usually involves data compression: representing the data with as few bits as possible. Notice that one always needs at least as many bits to encode a message as the entropy of the message. The next chapter talks more about source coding, and presents one special example: the Huffman code. 2. Channel Coding. Here one uses error detection and error correction to improve the reliability of the channel.